But again, Leyton was presumably very high up the Starfleet food chain, possibly even the chief of staff since he's the one meeting with the President on a regular basis and he doesn't answer to any other Starfleet admiral at all. If he were chief of staff, then there shouldn't be anything in Starfleet that would be classified from him.
Even if someone is very high up in Starfleet, it doesn't mean they would have access to ALL information even if he's a chief of staff.
You think Leyton would have access to Section 31 files and all their activities? Not really, because they don't answer to anyone... literally.
He'd only know about S31 if they wanted him to know about them.
Bashir also thought he had highest medical clearance in all of Starfleet, and he couldn't get files on Odo's infection for the life of him.
It wouldn't be that simple at all. After seizing power, there would very likely be planets in the Federation who would refuse to acknowledge the legitimacy of a government run by a Starfleet Admiral who illegally removed the sitting President from office, and therefore they aren't going to do what he orders them to just because he's in charge. A show of force would be necessary to get those planets in line.
Most of UFP would generally recognize whatever power is in place if there's a solid argument for it (in ST VI, there was literally a massive conspiracy in UFP government that involved multiple government parties which would have proceeded if it hadn't been exposed by Kirk)... and I suspect that Leyton would have thrown in a convincing argument for upgrading all of UFP's worlds orbital defenses and ships in service (if that was his goal)... its not an unreasonable suggestion after all (its a proposal that effectively keeps all of UFP up to date which increases protection of UFP member planets, and it helps to increase overall security and wellbeing of SF officers and civilians who explore space).
Leyton's issue was the use of force and extremism (marshal law and forced blood screenings) which affected the quality of life of people and turned it to fear... that's actually what backfired for him and resulted in his removal from power... which also happened because of Sisko and Odo exposing his criminal activities in the first place (and just prior to this exposure, Earth was already under Marshal Law and forced screenings however begrudgingly... and if he can do that on Earth... then why couldn't he do that on every other UFP member planet?)... if Leyton was successful and gotten there without that exposure (even if it took a bit longer originally), he probably would have gotten away with his plans across UFP anyway.
Anyway, my point is that Starfleet and UFP would have likely upgraded their defensive systems and active ships in the fleet well before the Dominion War (if they hadn't done so already) since the first Borg invasion (when Picard was assimilated)... having Betazed with 80 year old orbital defensive systems during the Dominion war was bonkers on the writers part and it literally makes 0 sense because it makes SF looks like bunch of idiots, and to be fair, no supposedly successful organization would have lasted as long with such poor decision making.